We deal with the problem of extracting as much randomness as possible from a defective random source. We devise a new tool, a ``merger'', which is a function that accepts d strings, one of which is uniformly distributed, and outputs a single string that is guaranteed ...
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We consider the existence of pairs of probability ensembles whichmay be efficiently distinguished given $k$ samples but cannot be efficiently distinguished given $k'<k$ samples.It is well known that in any such pair of ensembles it cannot be thatboth are efficiently computable(and that such phenomena ...
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In this paper, we give explicit constructions of extractors which work fora source of any min-entropy on strings of length $n$. The firstconstruction extracts any constant fraction of the min-entropy usingO(log^2 n) additional random bits. The second extracts all themin-entropy using O(log^3 n) additional random ...
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We introduce a new approach to construct extractors -- combinatorial objects akin to expander graphs that have several applications. Our approach is based on error correcting codes and on the Nisan-Wigderson pseudorandom generator. An application of our approach yields a construction that is simple to ...
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We give explicit constructions of extractors which work for a source ofany min-entropy on strings of length n. These extractors can extract anyconstant fraction of the min-entropy using O(log^2 n) additional randombits, and can extract all the min-entropy using O(log^3 n) additionalrandom bits. Both of these ...
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We give the first construction of a pseudo-random generator withoptimal seed length that uses (essentially) arbitrary hardness.It builds on the novel recursive use of the NW-generator ina previous paper by the same authors, which produced many optimal generators one of which was pseudo-random. This is achieved ...
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Weak designs were defined by Raz, Reingold and Vadhan (1999) and areused in constructions of extractors. Roughly speaking, a weak designis a collection of subsets satisfying some near-disjointnessproperties. Constructions of weak designs with certain parameters aregiven in [RRV99]. These constructions are explicit in the sense thatmore >>>

On an input probability distribution with some (min-)entropy an {\em extractor} outputs a distribution with a (near) maximumentropy rate (namely the uniform distribution).A natural weakening of this concept is a condenser, whoseoutput distribution has a higher entropy rate than theinput distribution (without losingmuch of ...
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The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and ...
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Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, ...
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Mergers are functions that transform k (possibly dependent)random sources into a single random source, in a way that ensuresthat if one of the input sources has min-entropy rate $\delta$then the output has min-entropy rate close to $\delta$. Mergershave proven to be a very useful tool in ...
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Optimal dispersers have better dependence on the error thanoptimal extractors. In this paper we give explicit disperserconstructions that beat the best possible extractors in someparameters. Our constructions are not strong, but we show thathaving such explicit strong constructions implies a solutionto the Ramsey graph construction ...
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Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ...
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We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as ...
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In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science,probabilistic algorithms are sometimes simpler and more efficientthan the best known ...
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A number of recent results have constructed randomness extractorsand pseudorandom generators (PRGs) directly from certainerror-correcting codes. The underlying construction in theseresults amounts to picking a random index into the codeword andoutputting $m$ consecutive symbols (the codeword is obtained fromthe weak random source in the case ...
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In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial ...
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We study multilinear formulas, monotone arithmetic circuits, maximal-partition discrepancy, best-partition communication complexity and extractors constructions. We start by proving lower bounds for an explicit polynomial for the following three subclasses of syntactically multilinear arithmetic formulas over the field C and the set of variables {x1,...,xn}:

We give polynomial time computable extractors for low-weight affine sources. A distribution is affine if it samples a random point from some unknown low dimensional subspace of F^n_2 . A distribution is low weight affine if the corresponding linear space has a basis of low-weight vectors. Low-weight ane sources are ...
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An algebraic source is a random variable distributeduniformly over the set of common zeros of one or more multivariatepolynomials defined over a finite field $F$. Our main result isthe construction of an explicit deterministic extractor foralgebraic sources over exponentially large prime fields. Moreprecisely, we give ...
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A merger is a probabilistic procedure which extracts therandomness out of any (arbitrarily correlated) set of randomvariables, as long as one of them is uniform. Our main result isan efficient, simple, optimal (to constant factors) merger, which,for $k$ random vairables on $n$ bits each, uses a ...
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Let $\F$ be the field of $q$ elements. An \emph{\afsext{n}{k}} is a mapping $D:\F^n\ar\B$such that for any $k$-dimensional affine subspace $X\subseteq \F^n$, $D(x)$ is an almost unbiasedbit when $x$ is chosen uniformly from $X$.Loosely speaking, the problem of explicitly constructing affine extractors gets harder as $q$ gets ...
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The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this problem ...
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We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that adistribution $X$ on binary strings of length $n$ is a$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$to any string of length $n$. For every $\delta>0$ we construct thefollowing poly($n$)-time ...
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Two-source and affine extractors and dispersers are fundamental objects studied in the context of derandomization. This paper shows how to construct two-source extractors and dispersers for arbitrarily small min-entropy rate in a black-box manner from affine extractors with sufficiently good parameters. Our analysis relies on the study of approximate duality, ...
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Let $F$ be the field of $q$ elements, where $q=p^{\ell}$ for prime $p$. Informally speaking, a polynomial source is a distribution over $F^n$ sampled by low degree multivariate polynomials. In this paper, we construct extractors for polynomial sources over fields of constant size $q$ assuming $p \ll q$.

A $c$-short program for a string $x$ is a description of $x$ of length at most $C(x) + c$, where $C(x)$ is the Kolmogorov complexity of $x$. We show that there exists a randomized algorithm that constructs a list of $n$ elements that contains a $O(\log n)$-short program for $x$. ...
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We introduce and construct a pseudorandom object which we call a local correlation breaker (LCB). Informally speaking, an LCB is a function that gets as input a sequence of $r$ (arbitrarily correlated) random variables and an independent weak-source. The output of the LCB is a sequence of $r$ random variables ...
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A circuit $C$ \emph{compresses} a function $f:\{0,1\}^n\rightarrow \{0,1\}^m$ if given an input $x\in \{0,1\}^n$ the circuit $C$ can shrink $x$ to a shorter $\ell$-bit string $x'$ such that later, a computationally-unbounded solver $D$ will be able to compute $f(x)$ based on $x'$. In this paper we study the existence of ...
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A Boolean function on n variables is q-resilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining n-q variables; in other words, no coalition of at most q variables has significant influence on the ...
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We study how to extract randomness from a $C$-interleaved source, that is, a source comprised of $C$ independent sources whose bits or symbols are interleaved. We describe a simple approach for constructing such extractors that yields:

(1) For some $\delta>0, c > 0$,explicit extractors for $2$-interleaved sources on $\{ ...
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A non-malleable extractor is a seeded extractor with a very strong guarantee - the output of a non-malleable extractor obtained using a typical seed is close to uniform even conditioned on the output obtained using any other seed. The first contribution of this paper consists of two new and improved ...
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The main contribution of this work is an explicit construction of extractors for near logarithmic min-entropy. For any $\delta > 0$ we construct an extractor for $O(1/\delta)$ $n$-bit sources with min-entropy $(\log{n})^{1+\delta}$. This is most interesting when $\delta$ is set to a small constant, though the result also yields an ...
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We construct non-malleable extractors with seed length $d = O(\log{n}+\log^{3}(1/\epsilon))$ for $n$-bit sources with min-entropy $k = \Omega(d)$, where $\epsilon$ is the error guarantee. In particular, the seed length is logarithmic in $n$ for $\epsilon> 2^{-(\log{n})^{1/3}}$. This improves upon existing constructions that either require super-logarithmic seed length even for constant ...
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A typical obstacle one faces when constructing pseudorandom objects is undesired correlations between random variables. Identifying this obstacle and constructing certain types of "correlation breakers" was central for recent exciting advances in the construction of multi-source and non-malleable extractors. One instantiation of correlation breakers is correlation breakers with advice. These ...
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Our result extends the breakthrough result of Chattopadhyay and Zuckerman \cite{CZ15} and uses the non-malleable extractor of Cohen \cite{Cohen16}. The main new ingredient in our construction ...
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Let $\mathcal{F}$ be a finite alphabet and $\mathcal{D}$ be a finite set of distributions over $\mathcal{F}$. A Generalized Santha-Vazirani (GSV) source of type $(\mathcal{F}, \mathcal{D})$, introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence $(F_1, \dots, F_n)$ in $\mathcal{F}^n$, where $F_i$ is a sample from ...
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In their seminal work, Chattopadhyay and Zuckerman (STOC'16) constructed a two-source extractor with error $\varepsilon$ for $n$-bit sources having min-entropy $poly\log(n/\varepsilon)$. Unfortunately, the construction running-time is $poly(n/\varepsilon)$, which means that with polynomial-time constructions, only polynomially-large errors are possible. Our main result is a $poly(n,\log(1/\varepsilon))$-time computable two-source condenser. For any $k ...
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We study the task of seedless randomness extraction from recognizable sources, which are uniform distributions over sets of the form {x : f(x) = v} for functions f in some specified class C. We give two simple methods for constructing seedless extractors for C-recognizable sources.Our first method shows that ...
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